THE SORITES AND

LIU, YANG

Abstract. This paper studies degree theoretic approach to . The goal is to provide an explanation for the Sorites paradox from degree the- orists' perspective. The paper includes a discussion of degrees and tol- erance, both of which are taken to be ineliminable features of language governing the use of vague predicates. A defense of degree theory in re- sponse to some challenges in the literature is inserted in its due place. The will then lead to the introduction of a basic propositional fuzzy logic which will serve as a conceptual framework within which the Sorites are treated. The paper shows that there is a way of treating toler- ance within degree theory by introducing a fuzzy notion of validity.

1. Introduction 1.1. Soros. Theories of vagueness are often motivated by solving the an- cient riddle of heap (Greek: soros): a heap of sand remains a heap after the removal of one grain of sand, but will cease to be a heap if a large amount of sand is removed. An immediate lesson we learn from the riddle is that a significant change can be the result of joining small ones which, if viewed individually, do not seem to make any difference. Riddles which exemplify the same pattern of reasoning as the soros are known in the literature as the "Sorites ." 1.2. Sorites chain. For further discussion, it is customary to put a sorites in a more transparent form as follows. Let P be a sorites (e.g. ``heap'', ``bald'', etc.), and a0,..., an, (n < ∞) a sequence of objects predicated on P. Then the argument consists in the following Sorites chain: → Pa0 Pa0 Pa1 → Pa1 Pa1 Pa2 Pa2 . .. → Pan−1 Pan−1 Pan Pan → where each intermedia step Pai Pai Pai+1 (i < n) is an instantiation of modus Pa + →i 1 ponens, and, for each i, call Pai Pai+1 a Sorites conditional.

This paper was prepared for Professor Haim Gaifman's Seminar on Vagueness, 2010. 1 2 LIU, YANG

The ``paradox'' arises once we note that, on the one hand, each Sorites chain is a finite series of , and hence deductively valid, the truth of is supposed to be preserved in a deductively valid argu- ment; yet, on the other hand, a0 clearly falls under P, an clearly does not, and at each intermedia step if we agree that a small increment would not change our opinion on how objects are predicated then if ai falls under P so does ai+1, however if we join all the intermedia steps together, we can deduce from a0 that an falls under P, a contradiction.

1.3. Responses. The predicates ``heap'', ``bald'', ``child'', ``red'', ``small'', ``noonish'', and ``walking distance'' are paradigmatically vague. Various theories of vagueness have been proposed and studied in the last forty years. Most claim that vagueness involves borderline cases (the idea of lack- ing ``sharp boundaries'' of partitioning class of objects to which a vague predicate applies, objects in the intermediate steps of a Sorites chain are of- ten examples of borderline cases); and the phenomena of vagueness can be explained in terms of some kind of ignorance (the state of lacking certain factual knowledge or adequate semantic mechanism). To put in a concise form, let P be a monadic vague predicate and a be a borderline case of P, then a competent language user might vacillate between Pa and ¬Pa. The competing accounts of vagueness then dispute the nature of our ig- norance (or, to put in a milder term, our embarrassment about where to ``draw the line'' among borderline cases). The epistemic thesis has bitten the bullet and accepted that there are sharp cut-off points: a heap of sand will cease to be a heap after the removal of one grain of sand; there exists one hair the loss of which will turn a man into a bald man; Mary will no longer be a child with exactly one more heartbeat. But these cut-off points, according to this approach, are to which we have no access. Our igno- rance is therefore due to our inability to grasp the precise boundaries to this factual knowledge. In other words, for epistemicists, the indeterminacy of an object falling under a vague predicate is an epistemic problem: each bor- derline case either does or does not fall under the vague predicate, but the exact case is unknown, or even unknowable, to human beings. In contrast to the epistemic concept of vagueness, advocates of super- valuationism treat vagueness as a linguistic phenomenon, where indeter- minacy of vague predicates is given a semantic analysis as opposed to an epistemic one. According to this account, the truth of one object classified under a vague predicate or its negation is identified with supertruth, that is, true in every ``sharpening'' or ``precisification''. The idea is that a vague predicate can be sharpened by giving a precise interpretation such that the vagueness of the term is eliminated in the sharpening. There can be several precisifications of a vague predicate, but each admissible one shall satisfy certain semantic rules, for instance, if Mary is young then she was young THE SORITES PARADOX AND FUZZY LOGIC 3 a few seconds ago. A sentence is said to be supertrue (superfalse) if it is true (false) in every admissible precisification. Supervaluation semantics therefore admits truth-value gap: a sentence is neither true nor false if it is verified in some precisifications, but falsified in the others. For supervalu- ationists, the reason why we do not know the truth-value of Pa is because there is nothing to know (it is neither true nor false): the semantic defi- ciency facilitates our ignorance. In this way sorties paradoxes are said to be defused, because, although each Sorites conditional in penumbra is neither true nor false, the conjunction of all conditionals is superfalse (since in each sharpening there exists a conjunct which is falsified in that sharpening). The Sorites argument is therefore (deductively) valid but unsound. Proponents of contexualism concur in the assessment of vagueness as an apparent semantic phenomenon, the paradoxical status of which arises from features of our use of language which is best characterized as context- dependent. This approach seeks to account for the lack of sharp boundaries in the extension of vague predicates by offering an explanation that our un- derstanding of vague terms shifts along with the contexts in which they are properly applied. In dealing with sorites argument, confronted with a pair of objects that are close to one another on a sorites chain, the context requires the vague term be always interpreted in such a way as to not distin- guish these adjacent items. If John is not a bald man with 40,000 hairs, then he will stay non-bald with one hair removed: the predicate ``non-bald,'' as used in this particular context, delineates implicitely a way to use the term consistently. Vague predicates thus appear tolerant, that is to say, a given vague predicate tends to tolerate, in a proper context, marginal changes in relevant respects.1 So, according to contexualists, a sorites argument is both valid and sound at local level: the contextual variation in the interpretation of the vague predicate masks any relevant boundaries that may exist in the series; problems however arise if someone tries to string too many sorites conditionals.

1.4. Degree theory. The view defended in the present paper is a version of degree theory of vagueness. Degree theory traces our embarrassment about where to draw a dividing line on a sorites chain or among borderline cases to the incompleteness of truth values. The use of comparatives in natural language is a starting point for considering degrees. ``Today's rain is heavier than yesterday's rain.'' This sentence, according to degree theory, amounts to saying that it is truer to say that ``Today's rain is heavy'' than to say that ``Yesterday's rain is heavy.'' In practice, when faced with a vague predicate P, there is no denying that we are often limited to two alternatives: either

1 The concept of tolerance was first introduced by Wright (1975) and later adopted by con- textualism writers, see Gaifman (2010, Part I) and Shapiro (2006, §2.2). 4 LIU, YANG assert P or its negation. But the comparative use of the predicate, as ex- emplified above, seems to suggest that an adequate semantic analysis shall be capable of presenting the situation using a richer structure, rather than a simple twofold classification, the latter misses what is distinctive about a vague predicate whose application is indeed a matter of degree. For degree theorists, the that John is balder than Bill is reflected in `John is bald' having a higher truth degree than `Bill is bald'. It is therefore meaningful to talk about the degree of a falling under P. There are degrees of baldness, degrees of childishness, degrees of redness, etc. Degree theorists' slogan is that ``truth comes with degrees.''

Preview. Section 2 addresses the very idea of degrees of truth in the context of vagueness. The notion is taken to be a useful instrument which plays an explanatory role in understanding vague predicates. This leads us to a dis- cussion of semantic relations between tolerance and its refinement within degree theory. I will also attempt a defense of degree theory in response to some challenges in the literature. Section 3 provides a general formal setup within which the Sorites para- dox will be treated. The way it is treated is described in the next section. The goal is to develop a logic of vagueness that can serve as a conceptual framework in which the phenomenon of vagueness can be reconstructed and treated in a controlled manner. The logic apparatus employed here is called basic many-valued sentential logic (BL). This system is introduced and developed in Hájek (1998). I provide in this section a short summary. Degree theory has a quick and satisfactory response to the sorites para- dox. Our explanation depends on a fuzzy notion of validity. In more detail: let Σ be a finite list of premises of an argument, B its conclusion, we say that the argument is valid to degree ϵ if and only if for all feasible truth assign- ments2 { } min JAK | A ∈ Σ ⇒ JBK ≥ ϵ. (†) where `⇒ ' is the residuum function in use (introduced in section 3). Intu- itively, (†) means that although the truth degree of the conclusion might be less than the truth degree of the that contains the least truth, it is still meaningful to measure the validity of the argument to certain degree. And one can stipulate a level of acceptance (ϵ) to indicate the degree of va- lidity that can be accepted in a fuzzy reasoning. This will be the main theme for the last section of the paper.

Note. Given the technical setting of the winter school, I will try to balance the philosophical discussion and the technical part of the paper.

2 A truth assignment σ is said to be feasible if the assignment does not violate any of semantic norms governing the current use of language. THE SORITES PARADOX AND FUZZY LOGIC 5

References Cignoli, R., Esteva, F., Godo, L., and Torrens, A. (2000). Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Computing, 4(2):106--112. Edgington, D. (1999). Vagueness by degrees. In Keefe, R. and Smith, P., editors, Vagueness : a reader. MIT Press, Cambridge, Mass. Gaifman, H. (2010). Vagueness, tolerance and contextual logic. , 174(1):5--46. Hájek, P. (1998). Metamathematics of Fuzzy Logic. Trends in Logic. Kluwer, Dordrecht ; Boston. Keefe, R. and Smith, P. (1999). Vagueness : a reader. MIT Press, Cambridge, Mass. Priest, G. (2008). An Introduction to Non-classical Logic: From If to is. Cam- bridge Introductions to Philosophy. Cambridge University Press, 2 edi- tion. Shapiro, S. (2006). Vagueness in Context. Oxford University Press : Claren- don Press, Oxford; New York. Wright, C. (1975). On the coherence of vague predicates. Synthese, 30(3/4):325--365.

Department of Philosophy, Columbia University, New York, NY 10027 E-mail address: [email protected]